Grate the cheese. If needed, grate 8 ounces sharp cheddar cheese on the large holes of a box grater (about 2 cups).
Melt the butter. Melt 4 tablespoons unsalted butter in a medium saucepan over medium heat.
Add the flour. Add 2 tablespoons all-purpose flour and immediately stir with a wooden spoon, taking care to scrape the corners, until the flour is completely absorbed, 30 seconds to 1 minute. The mixture will be foamy and there should be no dry pockets of flour left.
Add the milk. While whisking constantly, slowly pour in 1 1/2 cups milk. Continue cooking over medium heat, whisking often, until the mixture comes to a simmer, the flour has completely dissolved, and the milk has thickened enough to coat the back of a spoon, 2 to 3 minutes.
Add the seasonings and cheese. Add 1/2 teaspoon kosher salt, 1/2 teaspoon paprika, and 1/4 teaspoon black pepper, and whisk to combine. Reduce the heat to low. Add the cheese a handful at a time, whisking between each addition, until the cheese is fully melted before adding more.